U.S. patent application number 17/190445 was filed with the patent office on 2022-05-05 for space-time fractional conductivity modeling of two-phase conducting media and simulation method thereof.
The applicant listed for this patent is Jilin University. Invention is credited to Shanshan GUAN, Yanju JI, Dongsheng LI, Jun LIN, Hui LUAN, Shipeng WANG, Yuan WANG, Qiong WU, Yibing YU, Xuejiao ZHAO.
Application Number | 20220137252 17/190445 |
Document ID | / |
Family ID | 1000005481217 |
Filed Date | 2022-05-05 |
United States Patent
Application |
20220137252 |
Kind Code |
A1 |
JI; Yanju ; et al. |
May 5, 2022 |
SPACE-TIME FRACTIONAL CONDUCTIVITY MODELING OF TWO-PHASE CONDUCTING
MEDIA AND SIMULATION METHOD THEREOF
Abstract
Provided is a space-time fractional conductivity modeling and
simulation method of two-phase conducting media, including: 1)
setting a simulated computation area, setting electric field or
magnetic field distribution nodes in the simulated computation
area, and setting an artificial current source at the origin of
coordinates; 2) selecting a shape function in the entire
computation area by a meshless method, and setting shape function
parameters, Gaussian integral parameters, electromagnetic
parameters, distance between the transmitting system and the
receiving system, and the range of the frozen soil layer; 3)
loading a first computation point and searching for nodes in the
radius of the support domain, discretizing the definite integral by
a 4-point Gaussian integral equation, then interpolating and
summing to obtain the fractional derivative of the shape function,
assigning the shape function result to the corresponding position
of the large sparse matrix in the spatial fractional electric field
diffusion equation.
Inventors: |
JI; Yanju; (Changchun,
CN) ; ZHAO; Xuejiao; (Changchun, CN) ; YU;
Yibing; (Changchun, CN) ; WANG; Shipeng;
(Changchun, CN) ; LIN; Jun; (Changchun, CN)
; WU; Qiong; (Changchun, CN) ; LI; Dongsheng;
(Changchun, CN) ; LUAN; Hui; (Changchun, CN)
; WANG; Yuan; (Changchun, CN) ; GUAN;
Shanshan; (Changchun, CN) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Jilin University |
Changchun |
|
CN |
|
|
Family ID: |
1000005481217 |
Appl. No.: |
17/190445 |
Filed: |
March 3, 2021 |
Current U.S.
Class: |
702/57 |
Current CPC
Class: |
G01V 99/005 20130101;
G01R 33/10 20130101; G01V 3/38 20130101; G01R 29/0892 20130101 |
International
Class: |
G01V 3/38 20060101
G01V003/38; G01R 29/08 20060101 G01R029/08; G01R 33/10 20060101
G01R033/10; G01V 99/00 20060101 G01V099/00 |
Foreign Application Data
Date |
Code |
Application Number |
Oct 29, 2020 |
CN |
202011177685.9 |
Oct 29, 2020 |
CN |
202011177721.1 |
Claims
1. A method, comprising: 1) setting a simulated computation area,
setting electric field or magnetic field distribution nodes in the
simulated computation area, and setting an artificial current
source at an origin of coordinates; 2) selecting a shape function
in the simulated computation area by a meshless method, and setting
shape function parameters, Gaussian integral parameters,
electromagnetic parameters, distance between a transmitting system
and a receiving system, and a range of a frozen soil layer; 3)
loading a first computation point and searching for nodes in a
radius of a support domain, discretizing a definite integral by a
4-point Gaussian integral equation, then interpolating and summing
to obtain a fractional derivative of a shape function, assigning a
shape function result to a corresponding position of a large sparse
matrix in a spatial fractional electric field diffusion equation,
selecting a next computation point until all computation points are
processed to form a linear equation system for all nodes; 4)
applying Dirichlet boundary conditions at a boundary of the
computation area and selecting a frequency for an artificial
current source, and solving the linear equation system by a LU
decomposition method to obtain an electric field value at each node
and obtain a magnetic field value of a corresponding node by a curl
equation for an electric field; and 5) obtaining, by changing an
emission frequency, a distribution of electric field and magnetic
field values at different nodes and frequencies.
2. The method of claim 1, wherein the electromagnetic parameters of
numerical simulation comprise emission frequency, permeability,
dielectric constant, ground conductivity, air conductivity and
infinite frequency conductivity.
3. The method of claim 1, wherein 3) comprises: 31) establishing a
multi-scale space-time fractional conductivity model containing a
time fractional term and a space fractional term; 32) transforming,
by fractional operator transformation, a space fractional operator
in the multi-scale space-time fractional conductivity model into a
Laplacian operator of the electric field to obtain a fractional
Laplacian operator, to obtain the spatial fractional electric field
diffusion equation; 33) expanding, by a Caputo fractional
definition, the spatial fractional electric field diffusion
equation into a fractional differential form; 34) transforming, by
a radial point interpolation meshless method, a second-order
partial differential operation of the electric field into a
second-order partial differential interpolation of a shape
function, to complete a discretization of a differential term in a
Caputo fractional order; and 35) transforming, by a Gaussian
numerical integration method, an integral operation into Gaussian
numerical integration accumulation, to complete a discretization of
an integral term in the Caputo fractional order so that the spatial
fractional electric field diffusion equation is transformed into a
linear equation system about the electric field.
4. The method of claim 3, wherein in 31), the established
multi-scale space-time fractional conductivity model is expressed
by: .sigma. .function. ( .omega. ) = .sigma. 0 .function. ( i
.times. v ) .alpha. .times. ( 1 + f 1 .times. M 1 .function. [ 1 -
1 1 + ( i .times. .omega. .times. .tau. 1 ) C 1 ] + f 2 .times. M 2
.function. [ 1 - 1 1 + ( i .times. .omega. .times. .tau. 2 ) C 2 ]
) ( 1 ) ##EQU00021## in (1), .sigma.(.omega.) is a conductivity in
a frequency domain, i is an imaginary part, .omega. is an angular
frequency, .sigma..sub.0 is a value of a DC conductivity, f.sub.1
is a volume fraction of a type-l particle, M.sub.1 is a rock
material property tensor, .tau..sub.1 is a time constant of the
type-l particle, C.sub.1 is a dispersion coefficient of the type-l
particle, (i.nu.).sup..alpha. corresponds to a space fractional
derivative for a Fourier mapping, .nu. is a dimensionless geometric
factor, and .alpha. is a fractal dimension of an anomaly.
5. The method of claim 3, wherein in 32), the multi-scale
space-time fractional conductivity model expression (1) is
substituted into the diffusion equation of a frequency domain
electric field of two-phase conducting media: .gradient. 2 .times.
E - .sigma. 0 .function. ( i .times. v ) .alpha. .times. ( 1 + f 1
.times. M 1 .function. [ 1 - 1 1 + ( i .times. .omega. .times.
.tau. 1 ) C 1 ] + f 2 .times. M 2 .function. [ 1 - 1 1 + ( i
.times. .omega. .times. .tau. 2 ) C 2 ] ) .times. ( i .times.
.times. .omega. .times. .times. E ) = 0 ( 2 ) ##EQU00022## both
ends of equation (2) are multiplied by (i.nu.)-.sup..alpha.: (
.gradient. v 2 ) s .times. E - .sigma. 0 .function. ( 1 + f 1
.times. M 1 .function. [ 1 - 1 1 + ( i .times. .omega. .times.
.tau. 1 ) C 1 ] + f 2 .times. M 2 .function. [ 1 - 1 1 + ( i
.times. .times. .omega..tau. 2 ) C 2 ] ) .times. ( i .times.
.times. .omega. .times. .times. E ) = 0 ( 3 ) ##EQU00023## where s
= 1 - .alpha. 2 , ##EQU00024## (.gradient..sub..nu..sup.2).sup.2 is
the fractional Laplacian operator in dimensionless coordinates v;
the spatial fractional electric field diffusion equation is: (
.gradient. v 2 ) s .times. E = .differential. 2 .times. s .times. E
.differential. x 2 .times. s + .differential. 2 .times. s .times. E
.differential. y 2 .times. s + .differential. 2 .times. s .times. E
.differential. z 2 .times. s ( 4 ) ##EQU00025## where E represents
the electric field, and x, y, and z each represent a deflection of
the electric field in each direction.
6. The method of claim 5, wherein in 33), by the Caputo fractional
definition expansion, the space fractional differential term in
equation (4) is discretized and approximated: .differential. 2
.times. s .times. E .differential. u 2 .times. s = 1 .GAMMA.
.function. ( 2 - 2 .times. s ) .times. .intg. a u .times. E ( 2 )
.function. ( .tau. ) ( u - .tau. ) 2 .times. s - 1 .times. d
.times. .tau. + 1 .GAMMA. .function. ( 2 - 2 .times. s ) .times.
.intg. u b .times. E ( 2 ) .function. ( .tau. ) ( u - .tau. ) 2
.times. s - 1 .times. d .times. .times. .tau. ( 5 ) ##EQU00026##
where u=x, y or z, .GAMMA. is a gamma function, a is a lower limit
of integration in a u direction, b is an upper limit of integration
in the u direction, .tau. is an integral variable, and
.GAMMA.(.alpha.) is the gamma function.
7. The method of claim 6, wherein in 34), transforming, by a radial
basis function meshless method, the second-order partial
differential operation of the electric field into the second-order
partial differential interpolation of a shape function to complete
the discretization of the differential term in the Caputo
fractional order in Equation (5) comprises: .differential. 2
.times. s .times. E .differential. u 2 .times. s = i = 1 n .times.
[ 1 .GAMMA. .function. ( 2 - 2 .times. s ) .times. .intg. a u
.times. .PHI. ui ( 2 ) .function. ( .tau. ) ( u - .tau. ) 2 .times.
s - 1 .times. d .times. .tau. + 1 .GAMMA. .function. ( 2 - 2
.times. s ) .times. .intg. u b .times. .PHI. ui ( 2 ) .function. (
.tau. ) ( .tau. - u ) 2 .times. s - 1 .times. d .times. .tau. ]
.times. E i ( 6 ) ##EQU00027## where .GAMMA. is the gamma function,
E.sub.i is a number of interpolation nodes near E, .PHI..sub.ui is
a corresponding interpolation shape function, .PHI..sub.ui.sup.(2)
is an interpolation shape function used to find a second-order
partial derivative of u.
8. The method of claim 6, wherein in 35), transforming, by a
Gaussian numerical integration method, the integral operation into
Gaussian numerical integration accumulation to complete the
discretization of the integral term in the Caputo fractional order
comprises: transforming an integration interval into unit sub-units
by coordinate transformation, wherein, if .tau. = u - a 2 .times.
.eta. + u + a 2 , ##EQU00028## then: u - a 2 2 - 2 .times. s
.times. .intg. - 1 1 .times. .PHI. i ( 2 ) .function. ( u - a 2
.times. .eta. + u + a 2 ) ( u - u .times. .eta. + a .times. .eta. -
a ) 2 .times. s - 1 .times. d .times. .eta. ( 7 ) ##EQU00029##
discretizing the integral term by the Gaussian numerical
integration method: u - a 2 2 - 2 .times. s .times. k - 1 n .times.
A k .times. .PHI. i ( 2 ) .function. ( u - a 2 .times. .eta. k + u
+ a 2 ) ( u - u .times. .eta. k + a .times. .eta. k - a ) 2 .times.
s - 1 ( 8 ) ##EQU00030## where .eta..sub.k is a Gaussian
integration point and A.sub.k is a weight coefficient.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] Pursuant to 35 U.S.C. .sctn. 119 and the Paris Convention
Treaty, this application claims foreign priority to Chinese Patent
Application No. 202011177685.9 filed on Oct. 29, 2020, and to
Chinese Patent Application No. 202011177721.1 filed on Oct. 29,
2020. The contents of all of the aforementioned applications,
including any intervening amendments thereto, are incorporated
herein by reference. Inquiries from the public to applicants or
assignees concerning this document or the related applications
should be directed to: Matthias Scholl P. C., Attn.: Dr. Matthias
Scholl Esq., 245 First Street, 18th Floor, Cambridge, Mass.
02142.
BACKGROUND
[0002] The disclosure relates to a space-time fractional
conductivity modeling and simulation method of two-phase conducting
media, which is suitable for the three-dimensional simulation of
time domain electromagnetic multi-scale diffusion, especially for
the high-precision three-dimensional numerical simulation of the
induction and polarization effects caused by the complex geometric
structure of the actual earth media.
[0003] In time domain transient electromagnetic methods, a long
wire source or loop source is used to output time-varying current
underground to excite the earth media to generate an induced
electromagnetic field. By measuring electric or magnetic field
signals, the electrical differences and the structure of the
underground media are detected. As a non-uniform and strong
dissipative medium, the earth's lithology and physical properties
show high non-uniformity and non-linearity. Especially, resources
such as concealed or disseminated polymetallic deposits, oil and
gas reservoirs, composite oil and gas reservoirs, and geothermal
energy are all composite multi-phase conducting media, so
multi-scale measurement of complex physical features or parameters
becomes especially important. Low-resistance and high-polarization
anomalies are one of the important indicators for geophysical
methods to detect sulfide-type, lead-zinc-silver and other
polymetallic deposits, while high-resistance and high-polarization
anomalies are important indicators for identification of oil and
gas reservoirs. By the excitation of the alternating field, the
induction and polarization effects in the multi-phase conducting
media coexist and accompany each other. The induction response can
better distinguish the geological formation, and the polarization
response can effectively identify favorable oil and gas reservoirs
and metal mine anomalies.
[0004] At present, the research on the polarization effect in China
and abroad mainly focuses on the numerical calculation of the
electromagnetic response of complex polarization bodies in the
three-dimensional Cole-Cole model, and involves only the study of
electromagnetic single-scale diffusion. However, there has been no
relevant research on electromagnetic multi-scale diffusion. The
Cole-Cole or GEMTIP model can characterize only the induction and
polarization effects caused by the dispersion characteristics of
the media. For the induction effect caused by the geometric
structure in the oil and gas reservoirs and porous media, the
existing models can no longer accurately extract information about
resistivity.
SUMMARY
[0005] The disclosure relates to a space-time fractional
conductivity modeling and simulation method of two-phase conducting
media. A space fractional term is introduced into the conductivity
model of the two-phase conducting media to establish a multi-scale
space-time fractional conductivity model, in which the time
fractional term characterizes the porous polarization effect of the
media and the space fractional term characterizes the induction
response caused by the complex geometric structure of the media.
The newly constructed conductivity model is introduced into an
electromagnetic diffusion equation, and the time and space
fractional differential terms are solved in the frequency domain
using a combination of finite difference and meshless methods.
Finally, the numerical simulation of the time domain multi-scale
induction-polarization symbiosis effects of the electromagnetic
field is completed by frequency-time conversion.
[0006] A space-time fractional conductivity modeling and simulation
method of two-phase conducting media comprises:
[0007] 1) setting a simulated computation area, setting electric
field or magnetic field distribution nodes in the simulated
computation area, and setting an artificial current source at the
origin of coordinates;
[0008] 2) selecting a shape function in the entire computation area
by a meshless method, and setting shape function parameters,
Gaussian integral parameters, electromagnetic parameters, distance
between the transmitting system and the receiving system, and the
range of the frozen soil layer;
[0009] 3) loading a first computation point and searching for nodes
in the radius of the support domain, discretizing the definite
integral by a 4-point Gaussian integral equation, then
interpolating and summing to obtain the fractional derivative of
the shape function, assigning the shape function result to the
corresponding position of the large sparse matrix in the spatial
fractional electric field diffusion equation, selecting a next
computation point until all computation points are processed to
form a linear equation system for all nodes;
[0010] 4) applying Dirichlet boundary conditions at the boundary of
the computation area and selecting a frequency for the artificial
current source, and solving the linear equation system by a LU
decomposition method to obtain an electric field value at each node
and obtain the magnetic field value of a corresponding node by a
curl equation for the electric field; and
[0011] 5) obtaining, by changing the emission frequency, the
distribution of electric field and magnetic field values at
different nodes and frequencies.
[0012] In a class of this embodiment, the electromagnetic
parameters of numerical simulation comprise emission frequency,
permeability, dielectric constant, ground conductivity, air
conductivity and infinite frequency conductivity.
[0013] In a class of this embodiment, 3) comprises:
[0014] 31) establishing a multi-scale space-time fractional
conductivity model containing a time fractional term and a space
fractional term;
[0015] 32) transforming, by fractional operator transformation, the
space fractional operator in the multi-scale space-time fractional
conductivity model into a Laplacian operator of the electric field
to obtain a fractional Laplacian operator, to obtain a spatial
fractional electric field diffusion equation;
[0016] 33) expanding, by the Caputo fractional definition, the
spatial fractional electric field diffusion equation into a
fractional differential form;
[0017] 34) transforming, by a radial point interpolation meshless
method, the second-order partial differential operation of the
electric field into the second-order partial differential
interpolation of a shape function, to complete the discretization
of the differential term in the Caputo fractional order; and
[0018] 35) transforming, by a Gaussian numerical integration
method, the integral operation into Gaussian numerical integration
accumulation, to complete the discretization of the integral term
in the Caputo fractional order so that the spatial fractional
electric field diffusion equation is transformed into a linear
equation system about the electric field.
[0019] In a class of this embodiment, in 31), the established
multi-scale space-time fractional conductivity model is expressed
by:
.sigma. .function. ( .omega. ) = .sigma. 0 .function. ( i .times. v
) a .times. ( 1 + f 1 .times. M 1 .function. [ 1 - 1 1 + ( i
.times. .omega. .times. .tau. 1 ) C 1 ] + f 2 .times. M 2
.function. [ 1 - 1 1 + ( i .times. .omega. .times. .tau. 2 ) C 2 ]
) ( 1 ) ##EQU00001##
[0020] In (1), .sigma.(.omega.) is the conductivity in the
frequency domain, i is the imaginary part, .omega. is the angular
frequency, .sigma..sub.0 is the value of the DC conductivity,
f.sub.1 is the volume fraction of the type-l particle, M.sub.1 is
the rock material property tensor, .tau..sub.1 is the time constant
of the type-l particle, C.sub.1 is the dispersion coefficient of
the type-l particle, (i.nu.).sup..alpha. corresponds to the space
fractional derivative for the Fourier mapping, .nu. is the
dimensionless geometric factor, and .alpha. is the fractal
dimension of the anomaly.
[0021] In a class of this embodiment, in 32), the multi-scale
space-time fractional conductivity model expression (1) is
substituted into the diffusion equation of the frequency domain
electric field of the two-phase conducting media:
.gradient. 2 .times. E - .sigma. 0 .function. ( i .times. v ) a
.times. ( 1 + f 1 .times. M 1 .function. [ 1 - 1 1 + ( i .times.
.times. .omega..tau. 1 ) C 1 ] + f 2 .times. M 2 .function. [ 1 - 1
1 + ( i .times. .times. .omega..tau. 2 ) C 2 ] ) .times. ( i
.times. .times. .omega. .times. .times. E ) = 0 ( 2 )
##EQU00002##
[0022] Both ends of equation (2) are multiplied by
(i.nu.)-.sup..alpha.:
( .gradient. v 2 ) s .times. E - .sigma. 0 .function. ( 1 + f 1
.times. M 1 .function. [ 1 - 1 1 + ( i .times. .times. .omega..tau.
1 ) C 1 ] + f 2 .times. M 2 .function. [ 1 - 1 1 + ( i .times.
.times. .omega..tau. 2 ) C 2 ] ) .times. ( i .times. .times.
.omega. .times. .times. E ) = 0 ( 3 ) ##EQU00003##
[0023] where
s = 1 - .alpha. 2 , ##EQU00004##
(.gradient..sub..nu..sup.2).sup.s is the fractional Laplacian
operator in dimensionless coordinates .nu.; the spatial fractional
electric field diffusion equation is:
( .gradient. v 2 ) s .times. E = .differential. 2 .times. s .times.
E .differential. x 2 .times. s + .differential. 2 .times. s .times.
E .differential. y 2 .times. s + .differential. 2 .times. s .times.
E .differential. z 2 .times. s ; ( 4 ) ##EQU00005##
[0024] where E represents the electric field, and x, y, and z each
represent the deflection of the electric field in each
direction.
[0025] In a class of this embodiment, in 33), by the Caputo
fractional definition expansion, the space fractional differential
term in the equation (4) is discretized and approximated:
.differential. 2 .times. s .times. E .differential. u 2 .times. s =
1 .GAMMA. .function. ( 2 - 2 .times. s ) .times. .intg. a u .times.
E ( 2 ) .function. ( .tau. ) ( u - .tau. ) 2 .times. s - 1 .times.
d .times. .tau. + 1 .GAMMA. .function. ( 2 - 2 .times. s ) .times.
.intg. u b .times. E ( 2 ) .function. ( .tau. ) ( u - .tau. ) 2
.times. s - 1 .times. d .times. .times. .tau. ( 5 )
##EQU00006##
[0026] where u=x, y or z, .GAMMA. is the gamma function, a is the
lower limit of integration in the u direction, b is the upper limit
of integration in the u direction, .tau. is the integral variable,
and .GAMMA.(.alpha.) is the gamma function.
[0027] In a class of this embodiment, in 34), transforming, by a
radial basis function meshless method, the second-order partial
differential operation of the electric field into the second-order
partial differential interpolation of a shape function to complete
the discretization of the differential term in the Caputo
fractional order in Equation (5) comprises:
.differential. 2 .times. s .times. E .differential. u 2 .times. s =
i = 1 n .times. [ 1 .GAMMA. .function. ( 2 - 2 .times. s ) .times.
.intg. a u .times. .PHI. ui ( 2 ) .function. ( .tau. ) ( u - .tau.
) 2 .times. s - 1 .times. d .times. .tau. + 1 .GAMMA. .function. (
2 - 2 .times. s ) .times. .intg. u b .times. .PHI. ui ( 2 )
.function. ( .tau. ) ( .tau. - u ) 2 .times. s - 1 .times. d
.times. .tau. ] .times. E i ( 6 ) ##EQU00007##
[0028] where .GAMMA. is the gamma function, E.sub.i is a number of
interpolation nodes near E, .PHI..sub.ui is the corresponding
interpolation shape function, .PHI..sub.ui.sup.(2) is the
interpolation shape function used to find the second-order partial
derivative of u.
[0029] In a class of this embodiment, in 35), transforming, by a
Gaussian numerical integration method, the integral operation into
Gaussian numerical integration accumulation to complete the
discretization of the integral term in the Caputo fractional order
comprises:
[0030] first, transforming an integration interval into unit
sub-units by coordinate transformation, wherein, if
.tau. = u - a 2 .times. .eta. + u + a 2 , ##EQU00008##
then:
u - a 2 2 - 2 .times. s .times. .intg. - 1 1 .times. .PHI. i ( 2 )
.function. ( u - a 2 .times. .eta. + u + a 2 ) ( u - u .times.
.eta. + a .times. .eta. - a ) 2 .times. s - 1 .times. d .times.
.eta. ( 7 ) ##EQU00009##
[0031] then, discretizing the integral term by the Gaussian
numerical integration method:
u - a 2 2 - 2 .times. s .times. k = 1 n .times. A k .times. .PHI. i
( 2 ) .function. ( u - a 2 .times. .eta. k + u + a 2 ) ( u - u
.times. .eta. k + a .times. .eta. k - a ) 2 .times. s - 1 ( 8 )
##EQU00010##
[0032] where .eta..sub.k is the Gaussian integration point and
A.sub.k is the weight coefficient.
[0033] The disclosure also provides a device for geological
exploration, the device comprising:
[0034] a computer, configured to simulate the distribution of
electric and magnetic field values in different geological
structures, different transmitting parameters and receiving
distances, and different nodes and different frequencies;
[0035] a transient electromagnetic (TEM) detection system
comprising a transmitting system and a receiving system; the
transmitting system being configured, according to different
geological structure characteristics and detection targets, to set
the transmitting parameters and the receiving distance, based on
the transmitting parameters and the receiving distance
corresponding to the geological electric field value and magnetic
field value under different frequencies simulated by the computer,
and to transmit the current according to the transmitting
parameters, and the receiving system being configured to
synchronously collect the geological signal excited by the
transmitting system.
[0036] In another aspect, the disclosure provides a method for
setting of parameters of the device for geological exploration, the
method comprising:
[0037] molding a geological structure, and simulating the
distribution of electric and magnetic field values under different
transmitting parameters and receiving distance, different nodes and
different frequencies; and
[0038] according to the characteristics of geological structure and
target to be detected, determining the transmitting parameters and
receiving distance corresponding to the electric field value and
magnetic field value of geology under different frequencies, and
setting the transmitting parameters and receiving distance of TEM
detection system.
[0039] The following advantages of the disclosure are associated
with the method of the disclosure: a multi-scale space-time
fractional conductivity model for complex rock structures is
proposed, which can accurately describe the induction-polarization
symbiosis effects of complex geometric structures. The fractional
Laplacian operator is simplified by the Caputo fractional
definition, to overcome the difficulty in solving the space
fractional differential. Furthermore, the differential and integral
terms are discretized respectively by the radial point
interpolation meshless method and the Gaussian numerical
integration method, which avoids too complex process, and provides
a theoretical basis for the electromagnetic wave propagation
mechanisms of complex geological structures.
BRIEF DESCRIPTION OF THE DRAWINGS
[0040] FIG. 1 is a flowchart of a space-time fractional
conductivity modeling and simulation method of two-phase conducting
media;
[0041] FIG. 2 is a view comparing the numerical solution by the
meshless method and the Mittag-Leffler function as the analytical
solution, by taking a one-dimensional diffusion equation as an
example;
[0042] FIG. 3 shows the influence of the fractal dimension on the
received induced electromotive force; and
[0043] FIG. 4 shows the influence of the polarizability on the
received induced electromotive force.
DETAILED DESCRIPTION
[0044] To further illustrate the disclosure, embodiments detailing
a space-time fractional conductivity modeling and simulation method
of two-phase conducting media are described below. It should be
noted that the following embodiments are intended to describe and
not limit disclosure.
[0045] With reference to FIG. 1, a space-time fractional
conductivity modeling and simulation method of two-phase conducting
media comprises:
[0046] 1) setting a simulated computation area, setting electric
field or magnetic field distribution nodes in the simulated
computation area, and setting an artificial current source at the
origin of coordinates;
[0047] 2) selecting a shape function in the entire computation area
by a meshless method, and setting shape function parameters,
Gaussian integral parameters, electromagnetic parameters, distance
between the transmitting system and the receiving system, and the
range of the frozen soil layer;
[0048] 3) loading a first computation point and searching for nodes
in the radius of the support domain, discretizing the definite
integral by a 4-point Gaussian integral equation, then
interpolating and summing to obtain the fractional derivative of
the shape function, assigning the shape function result to the
corresponding position of the large sparse matrix in the spatial
fractional electric field diffusion equation, selecting a next
computation point until all computation points are processed to
form a linear equation system for all nodes;
[0049] 4) applying Dirichlet boundary conditions at the boundary of
the computation area and selecting a frequency for the artificial
current source, and solving the linear equation system by a LU
decomposition method to obtain an electric field value at each node
and obtain the magnetic field value of a corresponding node by a
curl equation for the electric field; and
[0050] 5) obtaining, by changing the emission frequency, the
distribution of electric field and magnetic field values at
different nodes and frequencies; completing the numerical
simulation of the time domain multi-scale induction-polarization
symbiosis effects of the electromagnetic field by frequency-time
conversion, saving data, plotting and analyzing the data.
[0051] 3) comprises:
[0052] introducing a space fractional term into the conductivity
model of the two-phase conducting media to establish a multi-scale
space-time fractional conductivity model, wherein a time fractional
term characterizes the multi-capacitance polarization effect of the
media and a space fractional term characterizes the induction
effect caused by the complex geometric structure;
[0053] transforming, by fractional operator transformation, the
space fractional operation of the conductivity into a fractional
Laplacian operator to obtain a space fractional electromagnetic
diffusion equation;
[0054] expanding the fractional Laplacian operator in 2) into a
fractional differential form, and discretizing the fractional
differential in space by a Caputo fractional derivative;
[0055] transforming, by a radial point interpolation meshless
method, the second-order partial differential operation of the
electric field into the second-order partial differential
interpolation of a shape function, to complete the discretization
of the differential term in the Caputo fractional order;
[0056] transforming, by a Gaussian numerical integration method,
the integral operation into Gaussian numerical integration
accumulation, to complete the discretization of the integral term
in the Caputo fractional order so that the fractional
electromagnetic diffusion equation is transformed into a linear
equation system about the electric field.
[0057] Specifically, the established multi-scale space-time
fractional conductivity model is expressed by:
.sigma. .function. ( .omega. ) = .sigma. 0 .function. ( i .times. v
) a .times. ( 1 + f 1 .times. M 1 .function. [ 1 - 1 1 + ( i
.times. .omega. .times. .tau. 1 ) C 1 ] + f 2 .times. M 2
.function. [ 1 - 1 1 + ( i .times. .omega. .times. .tau. 2 ) C 2 ]
) ( 1 ) ##EQU00011##
[0058] In (1), .sigma.(.omega.) is the conductivity in the
frequency domain, i is the imaginary part, .omega. is the angular
frequency, .sigma..sub.0 is the value of the DC conductivity,
f.sub.1 is the volume fraction of the type-l particle, M.sub.1 is
the rock material property tensor, .tau..sub.1 is the time constant
of the type-l particle, C.sub.1 is the dispersion coefficient of
the type-l particle, (i.nu.).sup..alpha. corresponds to the space
fractional derivative for the Fourier mapping, .nu. is the
dimensionless geometric factor, and .alpha. is the fractal
dimension of the anomaly.
[0059] The multi-scale space-time fractional conductivity model
expression (1) is substituted into the diffusion equation of the
frequency domain electric field of the two-phase conducting
media:
.gradient. 2 .times. E - .sigma. 0 .function. ( i .times. v ) a
.times. ( 1 + f 1 .times. M 1 .function. [ 1 - 1 1 + ( i .times.
.times. .omega..tau. 1 ) C 1 ] + f 2 .times. M 2 .function. [ 1 - 1
1 + ( i .times. .times. .omega..tau. 2 ) C 2 ] ) .times. ( i
.times. .times. .omega. .times. .times. E ) = 0 ( 2 )
##EQU00012##
[0060] Both ends of formula (2) are multiplied by
(i.nu.)-.sup..alpha.:
( .gradient. v 2 ) s .times. E - .sigma. 0 .function. ( 1 + f 1
.times. M 1 .function. [ 1 - 1 1 + ( i .times. .times. .omega..tau.
1 ) C 1 ] + f 2 .times. M 2 .function. [ 1 - 1 1 + ( i .times.
.times. .omega..tau. 2 ) C 2 ] ) .times. ( i .times. .times.
.omega. .times. .times. E ) = 0 ( 3 ) ##EQU00013##
where
s = 1 - .alpha. 2 , ##EQU00014##
(.gradient..sub..nu..sup.2).sup.2 is the fractional Laplacian
operator in dimensionless coordinates .nu.; the spatial fractional
electric field diffusion equation is:
( .gradient. v 2 ) s .times. E = .differential. 2 .times. s .times.
E .differential. x 2 .times. s + .differential. 2 .times. s .times.
E .differential. y 2 .times. s + .differential. 2 .times. s .times.
E .differential. z 2 .times. s ; ( 4 ) ##EQU00015##
where E represents the electric field, and x, y, and z each
represent the deflection of the electric field in each
direction.
[0061] By the Caputo fractional definition expansion, the space
fractional differential term in the equation (4) is discretized and
approximated:
.differential. 2 .times. s .times. E .differential. u 2 .times. s =
1 .GAMMA. .function. ( 2 - 2 .times. s ) .times. .intg. a u .times.
E ( 2 ) .function. ( .tau. ) ( u - .tau. ) 2 .times. s - 1 .times.
d .times. .tau. + 1 .GAMMA. .function. ( 2 - 2 .times. s ) .times.
.intg. u b .times. E ( 2 ) .function. ( .tau. ) ( u - .tau. ) 2
.times. s - 1 .times. d .times. .times. .tau. ( 5 )
##EQU00016##
where u=x, y or z, .GAMMA. is the gamma function, a is the lower
limit of integration in the u direction, b is the upper limit of
integration in the u direction, .tau. is the integral variable, and
.GAMMA.(.alpha.) is the gamma function.
[0062] Transforming, by a radial basis function meshless method,
the second-order partial differential operation of the electric
field into the second-order partial differential interpolation of a
shape function to complete the discretization of the differential
term in the Caputo fractional order comprises:
.differential. 2 .times. s .times. E .differential. u 2 .times. s =
i = 1 n .times. [ 1 .GAMMA. .function. ( 2 - 2 .times. s ) .times.
.intg. a u .times. .PHI. ui ( 2 ) .function. ( .tau. ) ( u - .tau.
) 2 .times. s - 1 .times. d .times. .tau. + 1 .GAMMA. .function. (
2 - 2 .times. s ) .times. .intg. u b .times. .PHI. ui ( 2 )
.function. ( .tau. ) ( .tau. - u ) 2 .times. s - 1 .times. d
.times. .tau. ] .times. E i ( 6 ) ##EQU00017##
where .GAMMA. is the gamma function, E.sub.i is a number of
interpolation nodes near E, .PHI..sub.ui is the corresponding
interpolation shape function, .PHI..sub.ui.sup.(2) is the
interpolation shape function used to find the second-order partial
derivative of u.
[0063] Transforming, by a Gaussian numerical integration method,
the integral operation into Gaussian numerical integration
accumulation to complete the discretization of the integral term in
the Caputo fractional order comprises:
[0064] first, transforming an integration interval into unit
sub-units by coordinate transformation, wherein, by taking equation
(6) as an example, if
.tau. = u - a 2 .times. .eta. + u + a 2 , ##EQU00018##
then:
u - a 2 2 - 2 .times. s .times. .intg. - 1 1 .times. .PHI. i ( 2 )
.function. ( u - a 2 .times. .eta. + u + a 2 ) ( u - u .times.
.eta. + a .times. .eta. - a ) 2 .times. s - 1 .times. d .times.
.eta. ( 7 ) ##EQU00019##
[0065] then, discretizing the integral term by the Gaussian
numerical integration method:
u - a 2 2 - 2 .times. s .times. k = 1 n .times. A k .times. .PHI. i
( 2 ) .function. ( u - a 2 .times. .eta. k + u + a 2 ) ( u - u
.times. .times. .eta. k + a .times. .times. .eta. k - a ) 2 .times.
s - 1 ( 8 ) ##EQU00020##
where .eta..sub.k is the Gaussian integration point and A.sub.k is
the weight coefficient.
[0066] The aforesaid method is implemented through a device for
geological exploration, the device comprising:
[0067] a computer configured to simulate the distribution of
electric and magnetic field values in different geological
structures, different transmitting parameters and receiving
distances, and different nodes and different frequencies; and
[0068] a transient electromagnetic (TEM) detection system
comprising a transmitting system and a receiving system; the
transmitting system being configured, according to different
geological structure characteristics and detection targets, to set
the transmitting parameters and the receiving distance, based on
the transmitting parameters and the receiving distance
corresponding to the geological electric field value and magnetic
field value under different frequencies simulated by the computer,
and to transmit the current according to the transmitting
parameters, and the receiving system being configured to
synchronously collect the geological signal excited by the
transmitting system.
[0069] When in use, the parameters of the TEM detection system are
set by the space-time fractional conductivity modeling and
simulation method of two-phase conducting media. The parameters
comprise transmitting parameters and receiving distance, and the
setting process comprises:
[0070] molding a geological structure, and simulating the
distribution of electric and magnetic field values under different
transmitting parameters and receiving distance, different nodes and
different frequencies; and
[0071] according to the characteristics of geological structure and
target to be detected, determining the transmitting parameters and
receiving distance corresponding to the electric field value and
magnetic field value of geology under different frequencies, and
setting the transmitting parameters and receiving distance of TEM
detection system.
EXAMPLE
[0072] With reference to FIG. 1, a space-time fractional
conductivity modeling and simulation method of two-phase conducting
media comprises:
[0073] 1) setting a computation area (x: -40 km to 40 km, and z:
-40 km to 40 km), in which total 101101=10201 nodes are uniformly
distributed with a spacing of 800 m; and applying Dirichlet
boundary conditions on four sides of the computation area, with an
artificial current source arranged at (0 m, 0 m)
[0074] 2) setting electromagnetic parameters in the entire
computation area: emission frequency of 2n Hz(n=0,1,2, . . . ,10),
permeability of 4.pi.*10.sup.-7, dielectric constant of
1/36.pi.*10.sup.-9, ground conductivity of 0.01 S/m, air
conductivity of 1*10.sup.-6 S/m, c of 0.5, time constant of 0.01 s,
infinite frequency conductivity of 0.1, frozen soil between 40 m
and 120 m, and sending and receiving distance of 20 m;
[0075] 3) setting parameters for the meshless method (including the
selection of shape function types and the setting of shape function
parameters and support domain parameters), initializing the large
sparse matrix K (10201.times.10201 in size), loading a first
computation point and searching for nodes in the radius of the
support domain, interpolating to obtain a shape function,
discretizing the definite integral by a 4-point Gaussian integral
equation, then interpolating and summing to obtain the fractional
derivative of the shape function, assigning the shape function
result to the corresponding position of the large sparse matrix,
selecting a next computation point from the nodes until all
computation points are processed to form a linear equation system
about the nodes, loading Dirichlet boundary conditions and a
current source, solving the linear equation system by a LU
decomposition method to obtain an electric field value at each
node, and by changing the current emission frequency, obtaining the
magnetic field values at different frequencies and then obtaining
the magnetic field values by the curl equation for the electric
field.
[0076] 4) completing the numerical simulation of the time domain
multi-scale induction-polarization symbiosis effects of the
electromagnetic field by frequency-time conversion, saving data,
and plotting. As shown in FIG. 2, the numerical solution by the
meshless method and the Mittag-Leffler function as the analytical
solution are basically consistent. As shown in FIG. 3, the fractal
dimension influences the amplitude of the received induced
electromotive force and the generation time of the opposite sign.
The greater the fractal dimension, the smaller the response
amplitude, and the earlier the generation of the opposite sign. As
shown in FIG. 4, the polarizability influences the generation time
of the opposite sign of the received induced electromotive force.
The greater the polarizability, the earlier the generation of the
opposite sign.
[0077] It will be obvious to those skilled in the art that changes
and modifications may be made, and therefore, the aim in the
appended claims is to cover all such changes and modifications.
* * * * *